The Trefftz finite elements modelling crack propagation
Abstract
Investigation of crack propagation can sometimes be a crucial stage of engineering analysis. The T–element method presented in this work is a convenient tool to deal with it. In general, T-elements are the Trefftz-type finite elements, which can model both continuous material and local cracks or inclusions. The authors propose a special T-element in a form of a pentagon with shape functions analytically modelling the vicinity of the crack tip. This relatively large finite element can be surrounded by even larger standard T-elements. This enables easy modification of the rough element grid while investigating the crack propagation. Numerical examples proved that the "moving pentagon" concept enables easy automatic generation of the T-element mesh, which facilitates observation of crack propagation even in very complicated structures with many possible crack initiators occurring for example in material fatigue phenomena.
Keywords
References
[1] T . Belytschko, T. Black. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering, 45: 601- 620, 1999.[2] S. Bordas. Extended Finite Element and Level Set Methods with Applications to Growth of Cracks and Biofilms. PhD thesis, Northwestern University Evanston, Illinois, December 2003.
[3] F. Erdogan, G.C. Sih. On the crack extension in plates under plane loading and transverse shear. Journal of Basic Engineering, Transactions of the ASME, 85D: 519- 527, 1963.
[4] J.A.T . Freitas, Z.Y. Ji. Hybrid-Trefftz equilibrium model for crack problems. International Journal for Numerical Methods in Engineering, 39: 569-584, 1996.
[5] J. Jirousek, N. Leon. A powerful finite element for plate bending. Computer Methods in Applied Mechanics and Engineering, 12: 77-96, 1977.
[6] J. Jirousek, A. Venkatesh. Hybrid-Trefftz plane elasticity elements with p-method capabilities. International Journal for Numerical Methods in Engineering, 35: 1443- 1472, 1992.
[7] J. Jirousek, A. Wróblewski. T-elements: State of the art and future trends. Archives of Computational Methods in Engineering, 3/4: 323-434, 1996.
[8] J . Jirousek, A.P. Zieliński. Survey of Trefftz-type element formulations. Computers and Structures, 63(2): 225- 242,1997.
[9] D.G. Lewicki, R. Ballarini. Effect of rim thickness on gear crack propagation path. NASA Technical Memorandum 107229, Army Research Laboratory Technical Report ARL-TR-1110. Prepared for the 7th International Power Transmission and Gearing Conference, October 6-9, 1996, sponsored by the ASME, San Diego, California. Available: http://gltrs.grc.nasa.gov / reports/1996/TM-I07229.pdf.
[10] N. Moes, J. Dolbow, T. Belytschko. A Finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46: 131- 150, 1999.
[11] R. Piltner. Special finite elements with holes and internal cracks. International Journal for Numerical Methods in Engineering, 21: 1471- 1485, 1985.
[12] Q.H. Qin. The Trefftz Finite and Boundary Element Method, WITpress, Southampton, 2000.
[13] H. Sanecki, A.P. Zieliński, S. Łaczek. Application of Analytical Finite Elements to Crack Problems (in Polish), Grant T11F 012 23 of KBN (Polish State Committee for Scientific Research), Report, Part I, Cracow, 2003.
[14] H. Sanecki, A.P. Zieliński. Crack propagation modelled by T-elements. Engineering Computations. 23(2): 100- 123,2006.
[15] H. Sanecki, A.P. Zieliński. Application of T-elements to crack path prediction in 2D engineering structures. CMM2005 Częstochowa 21-24 June 2005, pp. 27. ISBN 83-921605-7-6/CD.
[16] G.C. Sih. Energy-density concept in fracture mechanics. Engineering Fracture Mechanics, 5: 1037- 1040, 1973.
[17] G.C. Sih. Some basic problems in fracture mechanics and new concepts. Engineering Fracture Mechanics, 5: 365-377, 1973.
[18] G.C. Sih. Strain energy-density factor applied to mixed-mode crack problems. International Journal of Fracture, 10: 305- 321, 1974.
[19] B.A. Szabo, J. Babuska. Computation of the Amplitude of Stress Singular Terms for Cracks and Reentrant Corners. Report WU /CCM-86/1, Washington Univ. In St. Louis, February 1986.
[20] P. Tong, P.H. Pian, S.L. Lasry. A hybrid-element approach to crack problems in plane elasticity. International Journal for Numerical Methods in Engineering, 7: 297- 308, 1973.
[21] G. N. Wells. Discontinuous Modelling of Strain Localisation and Failure. PhD thesis (Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft), 200l.
[22] M.L. Williams. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics, 19: 526-528, 1952.
[23] A.P. Zieliński, O.C. Zienkiewicz. Generalized finite element analysis with T-complete boundary solution functions. International Journal for Numerical Methods in Engineering, 21: 509- 528, 1985.
[24] A.P. Zieliński, I. Herrera. Trefftz method: fitting boundary conditions. International Journal for Numerical Methods in Engineering, 24: 871- 891, 1987.
[25] A.P. Zieliński. On trial functions applied in the generalized Trefftz method. Advances in Engineering Software, 24: 147- 155, 1995.
Published
Jul 21, 2022
How to Cite
SANECKI, Henryk; ZIELIŃSKI, Andrzej P..
The Trefftz finite elements modelling crack propagation.
Computer Assisted Methods in Engineering and Science, [S.l.], v. 15, n. 3-4, p. 353-368, july 2022.
ISSN 2956-5839.
Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/742>. Date accessed: 23 dec. 2024.
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